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Properties

Label 1-317-317.112-r0-0-0
Degree $1$
Conductor $317$
Sign $0.971 - 0.236i$
Analytic cond. $1.47214$
Root an. cond. $1.47214$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.476 + 0.878i)2-s + (−0.949 − 0.312i)3-s + (−0.545 + 0.838i)4-s + (−0.987 − 0.158i)5-s + (−0.177 − 0.984i)6-s + (−0.961 − 0.274i)7-s + (−0.996 − 0.0794i)8-s + (0.804 + 0.594i)9-s + (−0.331 − 0.943i)10-s + (0.216 + 0.976i)11-s + (0.780 − 0.625i)12-s + (−0.971 + 0.236i)13-s + (−0.216 − 0.976i)14-s + (0.888 + 0.459i)15-s + (−0.405 − 0.914i)16-s + (0.936 − 0.350i)17-s + ⋯
L(s)  = 1  + (0.476 + 0.878i)2-s + (−0.949 − 0.312i)3-s + (−0.545 + 0.838i)4-s + (−0.987 − 0.158i)5-s + (−0.177 − 0.984i)6-s + (−0.961 − 0.274i)7-s + (−0.996 − 0.0794i)8-s + (0.804 + 0.594i)9-s + (−0.331 − 0.943i)10-s + (0.216 + 0.976i)11-s + (0.780 − 0.625i)12-s + (−0.971 + 0.236i)13-s + (−0.216 − 0.976i)14-s + (0.888 + 0.459i)15-s + (−0.405 − 0.914i)16-s + (0.936 − 0.350i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $0.971 - 0.236i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 0.971 - 0.236i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5472856348 - 0.06567609950i\)
\(L(\frac12)\) \(\approx\) \(0.5472856348 - 0.06567609950i\)
\(L(1)\) \(\approx\) \(0.6372262188 + 0.1867548248i\)
\(L(1)\) \(\approx\) \(0.6372262188 + 0.1867548248i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (0.476 + 0.878i)T \)
3 \( 1 + (-0.949 - 0.312i)T \)
5 \( 1 + (-0.987 - 0.158i)T \)
7 \( 1 + (-0.961 - 0.274i)T \)
11 \( 1 + (0.216 + 0.976i)T \)
13 \( 1 + (-0.971 + 0.236i)T \)
17 \( 1 + (0.936 - 0.350i)T \)
19 \( 1 + (0.177 - 0.984i)T \)
23 \( 1 + (0.368 - 0.929i)T \)
29 \( 1 + (0.961 - 0.274i)T \)
31 \( 1 + (0.578 - 0.815i)T \)
37 \( 1 + (-0.869 - 0.494i)T \)
41 \( 1 + (-0.293 - 0.955i)T \)
43 \( 1 + (0.641 + 0.767i)T \)
47 \( 1 + (-0.578 + 0.815i)T \)
53 \( 1 + (-0.177 + 0.984i)T \)
59 \( 1 + (0.641 - 0.767i)T \)
61 \( 1 + (-0.671 + 0.741i)T \)
67 \( 1 + (-0.0198 - 0.999i)T \)
71 \( 1 + (-0.888 + 0.459i)T \)
73 \( 1 + (-0.255 - 0.966i)T \)
79 \( 1 + (-0.545 - 0.838i)T \)
83 \( 1 + (-0.0992 - 0.995i)T \)
89 \( 1 + (-0.476 - 0.878i)T \)
97 \( 1 + (0.869 + 0.494i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.931111011530669169547587721980, −23.84243299629975183149764176808, −23.192288864142148564092258467144, −22.50467638218248259645211258252, −21.80316489922071832663554634580, −20.988198224549052257908236448223, −19.595847794629117454025377890774, −19.19671869090978319710696383043, −18.31586213491166819236694495086, −16.98362342278745886225581736040, −16.06397190000996640400082756924, −15.260056822416989108848144054, −14.2205879852428212858709428345, −12.853757483981071982190399260973, −12.096847419939482856603413802325, −11.62134617437930810150500129986, −10.41029020315176036374303367004, −9.86686428483485004752532670013, −8.51155120231479824978764950721, −6.94115256428971463859916802012, −5.86591321147234170758025985250, −4.990228553100266013681773424582, −3.67915036501814637110661368818, −3.13262765885984456698677408203, −1.03822595656727694780992996681, 0.43666870545502795613436574323, 2.88772178036255322879172888155, 4.33873082437247497953031826051, 4.87880948880108062563692848103, 6.259472054587914588879656645264, 7.14937958046857903973102817440, 7.59399239108151725630079689709, 9.15924542290091591035606270038, 10.24805977709535240119595298162, 11.77139212117947898241237676951, 12.33282492693041171190455065597, 13.00233346681259816284559246988, 14.28063911184201132138549125623, 15.415357949611854351140577061804, 16.0645306450114662393620320696, 16.892974950066425439328117791697, 17.549174378917736762369625051835, 18.79413858303142900832167538650, 19.55119897702016680117636192514, 20.844461170708483700357221686199, 22.17321351517031890267331337707, 22.737664647790842868801056225484, 23.23702942044461048948448143480, 24.14853979755378575416026423856, 24.84033508583784805153509680065

Graph of the $Z$-function along the critical line